Expert: Paul Klarreich - 10/4/2006

Question
The first derivative equals 0 at points p and q .
The second derivative equals 0 at points p and r.

Does this mean that p and r are possible inflection points and q is the only possible relative extrema?

Answer
Roger Calc I Asks in Category Calculus ...
 
Subject:  Point of Inflection
 
Question:  The first derivative equals 0 at points p and q .
The second derivative equals 0 at points p and r.

Does this mean that p and r are possible inflection points and q is the only possible relative extrema?
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Hi, Roger,

The answer to the first is yes.  p and r are possible inflection points, since f'' is zero there.

But to answer the second question, you have to confirm that  f(x) is a well-behaved function, meaning that  f(x) is continuous and differentiable everywhere.  It is possible to have relative extrema [that's the plural of EXTREMUM] where f'(x) is undefined.

But let's assume f(x) doesn't do that.  Also assume that when you say 'equals 0 at' you mean to say 'equals 0 ONLY at'.  [Picky, picky, these math teachers.]

Could p be a relative extremum?  yes, of course it could.  Just look at the graph of  y = x^4.  It looks like a parabola, but shaped a little different.  At x = 0, f' = 0 and f'' = 0, but (0,0) is certainly a minimum.

Could q be a relative extremum?  It should be.  f'(q) = 0 and f''(q) /= 0, our assumption.  So this is definitely an extreme, of type depending on the sign of f''.